Chapter 4, Problem 2PS

Relative Extrema

(a) Graph the fourth-degree polynomial $f(x)=a{x}^4-6x^2$ for $a=-3,-2,-1,0,1,2,$ and 3. For what values of the constant a does p have a relative minimum or relative maximum?

(b) Show that p has a relative maximum for all values of the constant a.

(c) Determine analytically the values of a for which p has a relative minimum.

(d) Let $(x,y)=(x,p(x))$ be a relative extremum of p. Show that (x, y) lies on the graph of $y=-3x^2$. Verify this result graphically by graphing $y=-3x^2$ together with the seven curves from part (a).